In the previous post, we defined local zeta functions and meromorphically extended them to the complex plane. We now work towards defining the global zeta function. Although local zeta functions are supposed to be local factors of the zeta function, we are going to allow multiplying local zeta functions \( \zeta _ v(f _ v, c _ v) \) under some restrictive conditions. But let us first develop some general theory. 4. Fourier analysis on the adeles# We now start investigating the global analogue o…
April 21, 2018Mathematics24 minutes
The main goal of this series of posts is to give a brief summary of Tate’s thesis [Tat67]. Given a number field \( K \) and a character \( \chi : \mathrm{Cl} _ \mathfrak{m}(K) \rightarrow S^1 \), one defines the Dirichlet \( L \)-function as \( \displaystyle L(s, \chi) = \sum _ {\mathfrak{a} \subseteq \mathcal{O} _ K}^{} \frac{\chi(\mathfrak{a})}{(N\mathfrak{a})^s} = \prod _ {\mathfrak{p} \subseteq \mathcal{O} _ K}^{} \frac{1}{1 - (N\mathfrak{p})^{-s}}, \) where we regard \( \chi(\mathfrak{a}) =…
April 14, 2018Mathematics21 minutes
Classically, the localization \( \mathcal{C} [\mathcal{W}^{-1}] \) of a (small) category \( \mathcal{C} \) at a subcategory \( \mathcal{W} \) is given by: the objects of \( \mathcal{C}[\mathcal{W}]^{-1} \) are objects of \( \mathcal{C} \), the morphisms \( \mathcal{C}[\mathcal{W}^{-1}] (X, Y) \) are diagrams \( \displaystyle X \rightarrow X _ 1 \xleftarrow{f _ 1} X _ 2 \rightarrow X _ 3 \leftarrow \cdots \rightarrow X _ {2n-1} \xleftarrow{f _ n} X _ {2n} \rightarrow Y \) with \( f _ i \in \math…
March 20, 2018Mathematics16 minutes
Earlier, we have computed the dimension of \( H^i(\mathbb{P} _ k^n, \mathscr{O} _ X(m)) \) for all \( i \) and \( m \). It turned out that it is \( 0 \) for all \( i \neq 0, n \), and at \( i = 0, n \), we had \( \displaystyle \dim _ k H^0(\mathbb{P} _ k^n, \mathscr{O} _ X(m)) \cong \dim _ k H^n(\mathbb{P} _ k^n, \mathscr{O} _ X(-n-1-m)). \) This phenomenon is an instance of Serre duality. To explain this in detail, we first need to know a bit about \( \mathrm{Ext} \). 12. \( \mathrm{Ext} \) and…
January 14, 2018Mathematics13 minutes
It’s quite awkward to be introducing coherent sheaves at this point, when the title of the paper is "Coherent algebraic sheaves". So far, we’ve mostly gotten away with quasi-coherent sheaves. Serre introduces the notion of coherent sheaves, which contain some idea of finite generation in addition to being an quasi-coherent sheaf. But we want the class of sheaves to be form an abelian category. Finitely generated modules over a ring generally don’t form an abelian category, so we need an alternat…
January 12, 2018Mathematics10 minutes
As we know, cohomology of sheaves on affine schemes is boring. The next simplest case we can consider is sheaves on projective schemes. 7. The \( \operatorname{Proj} \) construction# We can define projective space by gluing affine space together, but there is a more canonical construction. By a graded ring \( S \), we implicitly assume that \( S = \bigoplus _ {d \ge 0} S _ d \). Definition 1. The scheme \( \operatorname{Proj} S \) is defined as the following. A point in \( \operatorname{Proj} S…
January 11, 2018Mathematics18 minutes
In the last post, we have defined sheaf cohomology as derived functors of the global sections. We showed that if \( X \) is a Noetherian scheme, it doesn’t matter if we derive from the category of \( \mathscr{O} _ X \)-modules or from quasi-coherent \( \mathscr{O} _ X \)-modules. This allowed us to immediately show that quasi-coherent sheaves over affine Noetherian schemes are acyclic. But this is not good enough if we want to compute cohomology of sheaves over non-affine schemes. It is very har…
January 10, 2018Mathematics18 minutes
Recently I’ve been reading Serre’s Faisceaux Algébriques Cohérents [Ser55]. The main point of the article is to develop the theory of sheaf cohomology and prove some properties of cohomology of coherent sheaves. Serre also computes the cohomology some of coherent sheaves on projective space. I don’t really like Serre’s treatment of varieties because it’s not based on the language of schemes. (Well I guess the language of schemes didn’t exist back then.) So I am going to mix bits of Serre [Ser55]…
January 8, 2018Mathematics18 minutes
The ring of Witt vectors is a functor that takes in a ring \( R \) and constructs another ring \( W _ p(R) \). If we take a finite field \( \mathbb{F} _ q \) with \( q = p^n \), it outputs a ring \( W _ p(\mathbb{F} _ q) = \mathcal{O} _ K \) where \( K \) is the unique unramified extension of \( \mathbb{Q} _ p \) of degree \( n \). So in some sense, Witt vectors give a canonical construction of local fields with a specific residue field. The theory of Witt vectors as developed by Witt [Wit37]. L…
December 6, 2017Mathematics24 minutes
I’ve recently been reading Torus actions on symplectic manifolds [Aud04] by Michèle Audin. I’d like to write down some of the facts I learned from this book. The main object the book covers is a symplectic manifold with a torus action. Why look at torus actions, not Lie group actions? This is not a rhetoric question, unfortunately. Torus actions seem to be quite important in various places, e.g., the classification of semisimple Lie algebras (Cartan subalgebras), but I have no idea why they are …
August 17, 2017Mathematics18 minutes
